We study the symmetry-protected topological (SPT) orders for bosonic systems from an information-theoretic viewpoint. We show that with a proper choice of the onsite basis, the degenerate ground-state space of SPT orders (on a manifold with boundary) is a quantum error-correcting code with macroscopic classical distance, hence is stable against any local bit-flip errors. We show that this error-correcting property of the SPT orders has a natural connection to that of the symmetry-breaking orders, whose degenerate ground-state space is a classical error-correcting code with a macroscopic distance, providing a new angle for the hidden symmetry-breaking properties in SPT orders. We further propose new types of topological entanglement entropy that probe the SPT orders hidden in their symmetric ground states, which also signal the topological phase transitions protected by symmetry. Combined with the original definition of topological entanglement entropy that probes the “intrinsic topological orders”, and the recent proposed one that probes the symmetry-breaking orders, the set of different types of topological entanglement entropy may hence distinguish topological orders, SPT orders, and symmetry-breaking orders, which may be mixed up in a single system.
We study the symmetry-protected topological (SPT) orders for bosonic systems from an information-theoretic viewpoint. We show that with a proper choice of the onsite basis, the degenerate ground-state space of SPT orders (on a manifold with boundary) is a quantum error-correcting code with macroscopic classical distance, hence is stable against any local bit-flip errors. We show that this error-correcting property of the SPT orders has a natural connection to that of the symmetry-breaking orders, whose degenerate ground-state space is a classical error-correcting code with a macroscopic distance, providing a new angle for the hidden symmetry-breaking properties in SPT orders. We further propose new types of topological entanglement entropy that probe the SPT orders hidden in their symmetric ground states, which also signal the topological phase transitions protected by symmetry. Combined with the original definition of topological entanglement entropy that probes the “intrinsic topological orders”, and the recent proposed one that probes the symmetry-breaking orders, the set of different types of topological entanglement entropy may hence distinguish topological orders, SPT orders, and symmetry-breaking orders, which may be mixed up in a single system.
We discuss the symmetry-protected topological (SPT) orders for bosonic systems from an information-theoretic viewpoint. We show that with a proper choice of the onsite basis, the degenerate ground-state space of SPT orders (on a manifold with boundary) is a quantum error-correcting code with macroscopic classical distance, hence is stable against any local bit-flip errors. We show that this error-correcting property of the SPT orders has a natural connection to that of the symmetry-breaking orders, whose degenerate ground-state space is a classical error-correcting code with a macroscopic distance, providing a new angle for the hidden symmetry-breaking properties in SPT orders. We propose new types of topological entanglement entropy that probe the STP orders hidden in their symmetric ground states, which also signal the topological phase transitions protected by symmetry. Combined with the original definition of topological entanglement entropy that probes the 'intrinsic topological orders', and the recent proposed one that probes the symmetry-breaking orders, the set of different types of topological entanglement entropy may hence distinguish topological orders, SPT orders, and symmetry-breaking orders, which may be mixed up in a single system.
Gapped interfaces (and boundaries) of two-dimensional (2D) Abelian topological phases are shown to support a remarkably rich sequence of 1D symmetry-protected topological (SPT) states. We show that such interfaces can provide a physical interpretation for the corrections to the topological entanglement entropy of a 2D state with Abelian topological order found by J. Cano, T. L. Hughes, and M. Mulligan [Phys. Rev. B 92, 075104 (2015)]. The topological entanglement entropy decomposes as $\ensuremath{\gamma}={\ensuremath{\gamma}}_{a}+{\ensuremath{\gamma}}_{s}$, where ${\ensuremath{\gamma}}_{a}>0$ only depends on universal topological properties of the 2D state, while a correction ${\ensuremath{\gamma}}_{s}>0$ signals the emergence of the 1D SPT state that is produced by interactions along the entanglement cut and provides a direct measure of the stabilizing symmetry of the resulting SPT state. A correspondence is established between the possible values of ${\ensuremath{\gamma}}_{s}$ associated with a given interface---which is named the ``boundary topological entanglement sequence''---and classes of 1D SPT states. We show that symmetry-preserving domain walls along such 1D interfaces (or boundaries) generally host localized parafermion-like excitations that are stable to local symmetry-preserving perturbations.
Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain anti-unitary time reversal symmetry) can be labeled by the elements in H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: (G_H, G_{\Psi}, H^{1+d}[G_{\Psi}, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_{\Psi} the symmetry group of the ground states.
Since the discovery of phase transitions driven by topological defects, the classification of phases of matter has been significantly extended beyond Ginzburg and Landau's paradigm of spontaneous symmetry breaking (SSB). In particular, intrinsic and symmetry-protected topological (SPT) orders have been discovered in (mostly gapped) quantum many-body ground states. However, these are commonly viewed as zero-temperature phenomena, and their robustness in a gapless ground state or against thermal fluctuations remains challenging to tackle. Here we introduce an explicit construction for SPT-type states with hidden order associated with SSB: They feature (quasi-)long-range correlations along appropriate edges but short-range order in the bulk, ground state degeneracy associated with SSB, and nonlocal string order in the bulk. We apply our construction to predict two types of finite-temperature SPT transitions protected by 1-form symmetries in the Ising and BKT class, respectively, where the usual signs of criticality appear despite the absence of a diverging correlation length in the bulk. While the state featuring hidden Ising order is gapped, the other SPT state associated with the BKT-SPT transition has hidden U ( 1 ) or XY order and constitutes an intrinsically gapless SPT state, associated with a gapless Goldstone mode. Specifically, in this work we discuss spins with global Z 2 or U ( 1 ) symmetry coupled to link variables constituting a loop gas model with a 1-form symmetry. By mapping this system to an Ising-gauge theory, we demonstrate that one of the SPT phases we construct corresponds to the Higgs-SPT phase at T = 0 – which we show here to remain stable at finite temperature. Our work paves the way for a more systematic search for hidden order SPT phases, including in gapless systems, and raises the question if a natural (finite- T ) spin liquid candidate exists that realizes hidden order in the Higgs-SPT class.
Symmetry-protected topological (SPT) phases with gapless edge excitations have been shown to exist in strongly interacting bosonic/fermionic systems and it is highly desirable to identify practical systems realizing such phases through numerical simulation. A central question to be addressed is how to determine the SPT order in the system given the simulation result, while no local order parameter can be measured to distinguish the phases from a trivial one. In the tensor network approach to simulate strongly interacting systems, the quantum state renormalization algorithm has been shown to be effective in identifying intrinsic topological orders. Here, we show that a modified algorithm can identify SPT orders by extracting the symmetry-protected fixed point entanglement pattern in the ground-state wave function, which is essential for the existence of SPT order. The key to this approach is to add proper symmetry protection to the renormalization process. We demonstrate the effectiveness of this algorithm with examples of nontrivial SPT phases with internal symmetry in 1D and internal and translation symmetry in 2D.
In recent years, topological phases of matter have presented exciting new avenues to achieve scalable quantum computation. In this thesis, we investigate a class of quantum many-body spin models known as symmetry-protected topological (SPT) phases for use in quantum information processing and storage. We explore the fault-tolerant properties of SPT phases, and how they can be utilized in the design of a quantum computer. Of central importance in this thesis is the concept of quantum error-correction, which in addition to its importance in fault-tolerant quantum computation, is used to characterise the stability of topological phases at finite temperature. We begin with an introduction to quantum computation, quantum error correction, and topological phases of matter. We then focus on the fundamental question of whether symmetry-protected topological phases of matter can exist in thermal equilibrium; we prove that systems protected by global onsite symmetries cannot be ordered at nonzero temperature. Subsequently, we show that certain three-dimensional models with generalised higher-form symmetries can be thermally SPT ordered, and we relate this order to the ability to perform fault-tolerant measurement-based quantum computation. Following this, we assess feasibility of these phases as quantum memories, motivated by the fact that SPT phases in three dimensions can possess protected topological degrees of freedom on their boundary. We find that certain SPT ordered systems can be self-correcting, allowing quantum information to be stored for arbitrarily long times without requiring active error correction. Finally, we develop a framework to construct new schemes of fault-tolerant measurement-based quantum computation. As a notable example, we develop a cluster-state scheme that simulates the braiding and fusion of surface-code defects, offering novel alternative methods to achieve fault-tolerant universal quantum computation.